Oil & Gas Science and Technology – Rev. IFP, Vol. 57 (2002), No. 6, pp. 637-651Copyright © 2002, Éditions Technip

Water Saturation Computation from Laboratory, 3D Regression

G.M. Hamada1, M.N.J. Al-Awad1 and A.A. Alsughayer1

1 Petroleum Engineering Department, College of Engineering, King Saud University, Saudi Arabia, e-mail: ghamada@ksu.edu.sa - malawwad@ksu.edu.sa - asughayer@ksu.edu.sa

Résumé — Estimation de la saturation en eau au laboratoire par régression 3D— La déterminationprécise des réserves initiales d’un réservoir est très importante pour obtenir une bonne estimation de lacapacité de production d’un gisement d’hydrocarbures. La formule modifiée d’Archie (Sw = (a Rw/φm Rt)

1/n)constitue l’équation de base au calcul de la saturation en eau dans le sable ou l’utilisation d’un modèle desaturation pour le sable argileux. Une bonne évaluation de la saturation en eau dépend de ladétermination la plus précise possible des paramètres d’Archie a, m, n. Cet article presente une nouvelle technique destinée à déterminer les paramètres a, m, nd’Archie. Cettetechnique est basée sur le principe de régression en 3D. Elle surmonte le problème de l’incertitude desvaleurs de saturation en eau.Deux exemples de terrain sont présentés afin de bien tester les résultats de la technique de régression 3Det les comparer à trois autres techniques — méthode CAPE (Core Archie-Parameters Estimation)méthode conventionnelle et valeurs par défaut — concernant les valeurs des paramètres a, m and n de laformule d’Archie et de la saturation en eau.

Abstract — Water Saturation Computation from Laboratory, 3D Regression— An accuratedetermination of initial oil in place in the early life of reservoirs or an evaluation of a developedreservoir is required to well estimate the hydrocarbon volumes. Modified Archie’s formula (Sw = (a Rw/φm Rt)

1/n) is the basic equation to compute water saturation in clean formation or suitable shaly watersaturation model in shaly formation. The accuracy of water saturation value for given reservoirconditions depends on the accuracy of Archie’s parameters a, m and n. The terms of Archie’srelationship have been subjected to many laboratory investigations and even more speculations. Thereare many factors affecting porosity exponent, m, saturation exponent, n and tortuosity factor, a.Therefore, it is very difficult to fix Archie’s parameters and neglecting reservoir characteristics such asrock wettability, formation water salinity, permeability, porosity and fluids distribution.This paper presents a new technique to determine Archie’s parameters a, m and n. The developedtechnique is based on the concept of three dimensional-regression (3D) plot of water saturation,formation resistivity and porosity. This 3D technique provides simultaneous values of Archie’sparameters. Also, the 3D technique overcomes the uncertainty problems due to the separate use offormation resistivity factor-porosity and water saturation equations to get a, m and n parameters.Two field examples are given to show the applicability of the 3D technique in comparison with threeother techniques: – common values of Archie’s parameters, – conventional technique and – Core Archie-Parameter Estimation (CAPE) technique.

Oil & Gas Science and Technology – Rev. IFP, Vol. 57 (2002), No. 6

NOMENCLATURE

a = tortuosity factorm= cementation factorn = saturation exponentSw = water saturation, fractionVsh= shale volume, Ω⋅mRsh= shale resistivityGR= gamma ray reading, APIRt = resistivity of rock,Ω⋅mRw = resistivity of brine,Ω⋅mRo = resistivity of rock with Sw= 1.0, Ω⋅mIr = resistivity indexF = formation resistivity factorφ = formation porosity, fractionσSw= standard deviation in water saturation.

INTRODUCTION

In routine formation evaluation Archie’s parameters a, mandn are held constant for a given sample of a reservoir rock. Ineffect, this presumed constancy formulates the basis for thedetermination of hydrocarbon saturation from resistivitymeasurements for a particular lithology. An increasingnumber of cases are being encountered where the saturationexponent, n, has been observed to vary from the commonvalue of 2 in strongly water wet reservoir rocks to 25 instrongly oil wet reservoir rocks. Wettability effects becomeimportant when brine saturation lowered. Field experiencehas also shown that the cementation factor, m, and thetortuosity factor, a, depend on the petrophysical andmechanical properties of a rock [1-3].

Petroleum literature presents an exhaustive review of theresults determining Archie’s parameters and also water

saturation computation processes. In quantitative loginterpretation, accurate water saturation requires good valuesof Archie’s parameters. These parameters are used either inArchie’s saturation equation in clean formation or in a shalysand water saturation model in shaly formation.In this paper, the authors propose a new technique todetermine Archie’s parameters, three dimensionalregression (3D) technique which is based on the analyticalexpression of 3D plot of Rt /Rw versus Sw and φ. Watersaturation profiles are calculated using common values(1,2,2), conventional, CAPE and 3D techniques forselected two wells. In the case of clean sands, Archieformula was used to compute water saturation. In the caseof shaly sand sections three shaly sand models wereapplied and the chosen water saturation was an averagevalue of the three models used.

1 CALCULATION OF ARCHIE’S PARAMETERS

An exact computation of water saturation using Archie’sformula is based on an accurate values of Archie’sparameters a, m and n. In this study, 10 clean sand coresamples were selected for wells A and B. For each coresample, the electrical resistivity Ro at 100% water saturationand Rt at different water saturation percentages weremeasured at room temperature. The resistivity of simulatedbrine was fixed at 0.12 Ω⋅m. This resistivity valuecorresponds to formation water resistivity 0.054 Ω⋅m atreservoir temperature.

Table 1 shows the values of formation resistivity factorF, (Ro /Rw) and porosity for 10 core samples of well A andwell B. Table 2 contains the measured water saturation forwell A and well B. The average water saturation value forwell A was 30.5% and for well B was 12.25%. These

638

The comparison among the four techniques has shown that 3D technique provides an accurate andphysically meaningful way to get Archie’s parameters a, m and n for given core samples. Watersaturation profiles, using Archie’s parameters obtained from the four techniques, have been obtained forthe studied section in the wells. These profiles have shown a significant difference in water saturationvalues. This difference could be mainly attributed to the uncertainty level for Archie’s parameters fromeach technique. The effect of saturation exponent on the accuracy of water saturation computation wastested using Archie’s parameters derived from conventional technique and 3D technique in the two wells.

TABLE 1

Formation resistivity factor, F and porosity, φ for 10 core samples

Well A

F 17.8 14.4 10 11.3 10.9 9.5 22.7 16.8 10.2 11.5φ 28.2 31.3 37.4 35.1 35.9 38 0.25 0.29 36.9 34.5

Well B

F 35 11 12.7 34 14.5 12.7 10.5 9.1 9.5 6.4φ 14.2 26.7 24.6 14.3 23 25.4 27.2 30.2 28.5 35.6

GM Hamada et al./ Water Saturation Computation from Laboratory, 3D Regression 639

TABLE 2

Laboratory measured water saturation of 10 especially preserved core samples

Well A

Sample 1 2 3 4 5 6 7 8 9 10Sw 33 29.5 28.5 33 31.5 30.5 31 29 31 28

Average water saturation = 30.5%.

Well B

Sample 1 2 3 4 5 6 7 8 9 10Sw 13 10.5 12 11.5 13 12.5 10.5 12 13 14.5

Average water saturation = 12.25%.

TABLE 3

Electrical resistivity, Rt; resistivity index, Ir; fractional water saturation, Sw;and core sample porosity,φ for 6 core samples from well A and well B

Well A

Sample No. φ Sw Rt Ir n

1 28.2 100 2.14 1 1.87

83 3.09 1.44

63.9 4.9 2.29

56.9 5.92 2.77

28.3 22.2 10.4

2 31.3 100 1.73 1 2.04

83.6 2.47 1.43

41.2 10.57 6.11

30 19.37 11.2

21.7 38.93 22.5

3 37.4 100 1.2 1 2.08

93.6 1.38 1.15

81.7 1.82 1.52

25.6 19.08 15.9

4 35.1 100 1.36 1 2.23

75.5 2.42 1.78

67.2 3.06 2.25

27.6 18.9 13.9

12.9 90.98 66.9

5 35.9 100 1.31 1 2.06

93.3 1.52 1.16

72.8 2.49 1.9

27.3 18.47 14.1

16.3 53.7 41

6 38 100 1.14 1 2.37

87.1 1.58 1.39

78.6 2.02 1.77

27.3 24.74 21.7

Average saturation exponent, n = 2.1.

Oil & Gas Science and Technology – Rev. IFP, Vol. 57 (2002), No. 6640

TABLE 3 (continued)

Electrical resistivity,Rt; resistivity index, Ir; fractional water saturation, Sw;and core sample porosity, φ for 6 core samples from well A and well B

Well B

Sample No. φ Sw Rt Ir n

1 14.2 100 4.2 1 2.17

71.1 8.81 2.09

66.6 10.11 2.41

55.4 15.16 3.61

40.8 28.74 6.84

35.9 38.18 9.09

2 26.7 100 1.32 1 1.88

57 3.78 2.86

25.3 17.36 13.15

16.3 39.84 30.18

15.9 40.88 31

15 47.60 36.06

3 24.6 100 1.52 1 2.09

90.2 1.91 1.26

73.3 3.24 2.13

35.5 12.98 8.54

25 28.04 18.45

24.5 29.17 19.19

4 14.3 100 4.08 1 2.37

77.1 7.52 1.84

45.2 26.6 6.52

40 34.87 8.55

36.9 42.41 10.39

35.9 45.32 11.11

5 25.2 100 1.46 1 2.12

75.5 2.66 1.78

52.3 5.78 3.96

26.2 24.75 16.95

21.5 39 26.71

21.1 40 27.39

6 23 100 1.74 1 2.53

40.9 16.46 9.46

39.4 18.55 10.66

29.8 36.65 21.01

19.9 99 56.89

19.3 110 63.21

Average saturation exponent, n = 2.19.

GM Hamada et al./ Water Saturation Computation from Laboratory, 3D Regression

laboratory water saturation values have been used incalculating Archie’s parameters while the average valueswere being taken as references for the computed watersaturation values from logging data with the use Archie’sparameters by different techniques. Table 3 illustrates coresample resistivity, Rt at different water saturation, resisti-vity index, Ir , porosity, φ for 6 core samples from wells Aand B. Using the same specially preserved core samples,the water saturation has been measured in the laboratory.

1.2 Conventional Determination of a, m and n

In 1942, Archie proposed an empirical relationship betweenrock resistivity, Rt, with its porosity, φ, and water saturationSw.

Snw= a Rw/φmRt = Ro / Rt = 1/ Ir (1)

Other terms Ir, m,and n represent the resistivity index, thecementation factor and the saturation exponent. He has alsoshown experimentally that the resistivity of rock fullysaturated with brine, Ro, is related to the brine resistivity, Rw,by:

Ro = F Rw (2)

where F is formation resistivity factor. Archie’s formula (F=1/φm’) has been modified and tortousity factor, a, was addedto Archie’s formula [4]. It has taken the following form:

F = a/φm (3)

1.2.1 Conventional Determination of a and m

The conventional determination of a and m is based on Equation (3) and is rewritten as:

log F = loga – m logφ (4)

Plot of log F vs. log φ is used to determine a and m for thecore samples. The cementation factor m, is determined fromthe slope of the least square fit straight line of the plottedpoints, while the tortuosity factor is given from the interceptof the line where φ= 1. Note that, in this plot, only points ofSw = 1.0 is used. Archie’s parameters a and m weredetermined as 1.36 and 2.03 for well A and 0.95 and 1.85 forwell B.

1.2.2 Conventional Determination of n

The classical process to determine saturation exponent n isbased on Equation (1). This equation is rewritten as:

log Ir = – n logSw (5)

The bi-logarithmic plot of Ir versus Swgives a straight linewith negative slope n. Table 3 contains the saturationexponent value for each core sample: this value is determined

form the bi-logarithmic plot of I r and Sw for each coresample. Sometimes, data are plotted as log Rt vs. log Sw. Thisform is mathematically equivalent to the bi-logarithmic plotof Ir and Sw and provides the same value of n. The retainedsaturation exponent is the average values of the 6 coresamples measurements. The saturation exponent equals to2.1 for well A and 2.19 for well B. It is obvious that theconventional technique treats the determination of n as aseparate problem from a and m. This separation is notphysically correct, thereby; it induces an error in the value ofwater saturation using Equation (1).

1.3 Core Archie-Parameter Estimation (CAPE)

A data analysis approach is presented in [5] to determineArchie’s parameters m and n and optionally n fromstandard resistivity measurements on core samples. Theanalysis method, Core Archie-Parameters Estimation(CAPE) determines m and n and optionally a by mini-mising the error between computed water and measuredwater saturations. The mean square saturation error ε, isgiven by:

ε = ∑j ∑i [Swij – (aRwij/φim Rtij)

1/n]2 (6)

Where j = core index, i = index for each of the core jmeasurements, Swij = i th laboratory measured watersaturation for core j (fraction), Rtij = ith laboratory measuredresistivity for core j, Ω·m, and φj = core j porosity (fraction).Equation (6) calculates the minimum error between measuredcore water saturation and computed water saturation byArchie’s formula, by adjusting m, nand optionally a in theequation.

Table 4 illustrates typical results from CAPE and conven-tional methods and shows a, mand n values calculated withthe two methods for clean sandstone core samples. It isobvious that the values of a, mand n are different for a givenset of points.

TABLE 4

Archie’s parameters values from four techniques for well A and well B

Well A

Technique a m n

Common values 0.65 2.15 2

Conventional 1.36 2.03 2.1

CAPE 1.64 1.87 2.21

3D 1.43 1.97 2.15

Well B

Technique a m n

Common values 0.65 2.15 2

Conventional 0.95 1.85 2.19

CAPE 1.04 1.79 1.95

3D 0.92 1.87 1.87

641

Oil & Gas Science and Technology – Rev. IFP, Vol. 57 (2002), No. 6

1.4 Three-Dimensional Regression (3D)

In this technique it is contended that, so far as Archie’sparameters are concerned, the error in the water saturationvalue should be kept minimum. This is because the watersaturation quantity is desired and is physically a meaningfulquantity. Here, a method is developed to determine Archie’sparameters a, m and n using standard resistivitymeasurements on core samples [6-10].

Methodology: The basis of the 3D technique is to view Swin Archie’s formula Equation (1) as a variable in three-dimensional regression plot of Sw, Rw/Rt and φ. The 3Dtechnique determines Archie’s parameters a, m and n bysolving three simultaneous equations of Sw, Rw/Rt and φ.Equation (1) is rearranged after taking the logarithm of bothsides in the form:

log Rw / Rt = – log a + m logφ+ n log Sw (7)

The left hand side of Equation (7) is a dependent variableof the two independent variables Sw and φ. Equation (7) is anequation of a plane in 3D space of coordinates X, Yand Z(X = log φ, Y= log Sw and Z = log Rw/Rt). The intersection ofthis plane with the plane (X = 0.0) gives a staight line ofslope (m), with the plane (Y = 0.0) giving a straight line withslope (n) and with the plane (Z = 0.0) provides the value of(a) parameter.

For a given set of data for a core sample, we can obtainan equivalent set of variables X, Yand Z. Equation (7) willtake the following form for imeasurement points:

Zi = – A+ m Xi + n Yi (8)

After normalizing Equation (8) for N readings, thefollowing three simultaneous equations are resulted.

Σ Zi = – N A+ m ΣXi + n ΣXi (9)

Σ Xi Zi = – A N ΣXi + mΣ Xi2 + n Σ Xi Yi (10)

Σ Yi Zi = – A NΣ Yi + mΣ Xi Yi + n ΣYi2 (11)

The solution of Equations (9-11) provides the values ofArchie’s parameters a, mand n for one core sample. For jcore samples, running the same analysis produces averagevalues of Archie’s parameters. Figure 1 illustrates flowchart to compute Archie’s parameters by 3D technique fori measurements on j core samples. Table 4 shows Archie’sparameters calculated by 3D technique in addition to,CAPE, conventional and common values.

Note that the CAPE method is based on the idea that thetwo logarithmic plots of F versusφ and Sw versus Ir are notthe optimum way of handling the problem. The comparisonbetween the two techniques showed that CAPE might notappear as optimal as the conventional method. Instead, weare presenting another approach, the 3D technique. In thistechnique, water saturation is treated as an independentvariable in 3D plot of electrical resistivity vs. water saturationand porosity.

Figure 1

Flow chart to compute Archie’s parameters with 3Dtechnique.

642

Start

End

Input core dataφi, Swi, Rw and Rti

Calculate for all core measurements Log (φi) = Xi Log (Swi) = Yi Log (Rw/Rti) = Zi

ΣXi = 0.0 ΣYi = 0.0 ΣZi = 0.0ΣX

2i = 0.0, ΣXi ·Yi = 0.0, ΣXi ·Zi = 0.0,

ΣYi ·Zi = 0.0, ΣY 2i = 0.0

CalculateΣXi, ΣYi, ΣZi, ΣX

2i, ΣXi, ΣYi,

ΣXi, Zi, ΣYi, Zi, ΣY 2i,

ΣZi = - na + m ΣXi + n ΣYiΣXi ·Zi = - anΣXi + m ΣX

2i + n ΣYi ·Zi

ΣYi ·Zi = - anΣYi + m ΣXi ·Yi + n ΣY 2i,

Compute for the number of core samples and core measurements j = 1.0 i = 1.0

i = i + 1

i = j + 1

Output a, m, and n

Solving the simultaneous equations

All measurementsfor one core

No

No

Yes

Yes

All measurementsfor one core

GM Hamada et al./ Water Saturation Computation from Laboratory, 3D Regression

1.4.1 Assumptions

First, 3D technique assumes that Archie’s formula isapplicable to the examined core samples. Also, the coresamples represent the zone of interest. For shaly sandstone,Archie’s formula must be modified to account for thepresence of shale and its effect on resistivity measurements.The user is free to select the appropriate clay model, andconsequently, the shaly sand water saturation equation. Thesecond assumption might be difficult to satisfy since it isdealing with the accuracy of the laboratory measurementsunder reservoir conditions. The third assumption deals withthe concept of the 3D technique, this means that the usermust be acquainted with the basis and limitations of eachmethod before using it.

2 APPLICATIONS

Two field examples are used for applying the calculatedArchie’s parameters to compute water saturation. Figures 2and 3 are logging suites of resistivity, neutron, GR anddensity for wells A and B. GR was used to calculate shalevolume ( Vsh = 0.33 (22*GR – 1), when Vsh is ≤ 0.10, section isconsidered clean sand and when Vsh is ≥ 0.75, it is taken asshale section. Neutron and density logs readings have beencorrected from shale effect to give corrected porosity valuesfor using either in Archie’s formula or in shaly watersaturation model. The studied field examples produce fromclean sand sections and shaly sand sections. Figure 4 showsflow chart for a computer program used to compute watersaturation in clean and shaly sections with the use of Archie’s

643

Figure 2

Well logging data for well A.

SP (MV)CAL (IN)

GR (API)

- 200.0 0.0

6.0000 16.000

0.0 100.00

NPHI

RHOB (G/C3)

DRHO (G/C3)-.0500 .45000

.45000 -.1500

2.000 3.0000

MSFL (Ω·m)

ILD (Ω·m).2000 2000.0

.2000

8500

8600

8700

8800

8900

9000

2000.0

Oil & Gas Science and Technology – Rev. IFP, Vol. 57 (2002), No. 6

parameters from each technique: Conventional, CAPE and 3D and also common Archie’s parameters for sands (a = 0.65,m = 2.15 and n= 2).

In clean sand, water saturation is computed by Archie’sformula:

Sw = (aRw / φm Rt )1/n (12)

An accurate computation of water saturation in shalysand is subjected to many uncertain parameters. It isnecessary to integrate the information from several types of logs using different interpretation models and localexperience, in order to accurately determine the desiredformation properties. GR log is used to determine shalevolume that has been used to correct density and neutronporosity values from shale effect to give a corrected effectiveporosity used in water saturation computation. In the analysis

program, clean sand is assumed for Vsh ≤ 0.10 and useArchie’s formula to find Sw, shaly sand section is assumedfor 0.10 ≤Vsh ≤ 0.75 and use shaly sand water saturationmodel to find Sw and shale section is assumed for Vsh ≥ 0.75and neglect it. Water saturations have been computed forclean sections and shaly sand sections using Archie’sparameters calculated by conventional, CAPE, 3D techniquesand common values. Repeating the computer program andeach time with different Archie’s parameters a, mhas donethis and n values as shown in Figure 4. Considering the factthat water saturation has been computed in high shale volumesections as well as in low shale volume sections, we usedthree shale models and final water saturation value is taken asthe average value of the three shaly sand water saturationmodels. In effect, clays are dispersed within sand grains and,

644

SP (MV)CAL (IN)

GR (API)

- 200.0 0.0

6.0000 16.000

0.0 100.00

NPHI

RHOB (G/C3)

DRHO (G/C3)-.0500 .45000

.45000 -.1500

2.000 3.0000

MSFL (Ω·m)

ILD (Ω·m).2000 2000.0

.20009500

9600

9700

9800

9900

10 000

2000.0

Figure 3

Well logging data for well B.

GM Hamada et al./ Water Saturation Computation from Laboratory, 3D Regression 645

Start

Stop

Rmf, a, m, n, SP, Rsh, GRsh, φNsh,ρsh, ρf, ρm, GRcl, Temp.

Depth (i ), ρb(i ), GR(i ), φN(i ), Rt(i )

Find Rweq by SP logRw = 3.3 * Rweq

i = i + 1

Vsh 75%≥

Vsh 10%≥

IGR(i) =

Vsh(i) = 0.33 * (22*IGR(i) - 1)

GR(i) - GRmin

GRmax - GRmin

φNc(i) = φN(i) - Vsh (i) * φNsh

φDc(i) = - Vsh(i) *ρm - ρb(i)

ρm - ρf

ρsh - ρm

ρm - ρf

φC(i) -φNc (i)

2 + φDc (i)2

2

φD(i) =ρm - ρb(i)

ρm - ρf

φ(i) =φN(i)2 + φD(i)2

2

Swp(i) = [(I / Rt(i) )/ Vsh(i) / Rsh + φc(i) /(a * Rw) ](1 - Vsh / 2) m / 20.5 0.5 2 / n

SwD(i) = [I / Rt / Vsh / Rsh + φ (a * Rw) ]0.5 0.50.5m15 2 / n

SwI(i) = [(I / Rt(i) )/ + φm / a * Rw + 2 * Vsh /

(1 - Vsh / 2)

(a * Rw Rsh / m)2 + Vsh / Rsh]2*(1 - Vsh / 2) I / n

Sw(i) =a * Rw

φ(i) m * Rt

Use Archie's Equation

Depth (i ), Swav (i )

No

No

Yes

Yes

Figure 4

Flow chart to compute water saturation.

Oil & Gas Science and Technology – Rev. IFP, Vol. 57 (2002), No. 6

so, we selected three dispersed clay models. In fact that thereis no general shaly sand model can be applied in all modes ofshale distribution and also at level of shale volume. Forexample Doll model [11] often gives optimistic results, but invery shaly pay zone can be unrealistic and Poupon et al. [12]model and Hossin model [13] often become pessimistic inshaly pay sand located in transition zones and their appli-cation may condemn pay zones of commercial significance.

Consequently, it is recommended in these field examplesto compute water saturation by the three models; Poupon etal., Doll and Indonesia [14]. Final water saturation value willbe the average value of the calculated water saturationsderived from the three models. Water saturation equations ofthe three models are given as:

Doll model [11]:

SwD(i) = [1/Rt/Vsh1.5/Rsh

0.5+ φ0.5m/(aRw)0.5] 2/n (13)

Poupon et al. model [12]:

Swp(i) = [(1/Rt(i))/Vsh(i)(1-Vsh/2)/Rsh

0.5

+ φc(i)m/2 /(a*Rw)0.5]2/n (14)

Indonesia model [14]:

SwI(i) = [(1/Rt(i))/φm/aRw + 2*Vsh

(1-Vsh/2)/(a*Rw·Rsh/φm)2

+ Vsh2*(1-Vsh/2) /Rsh

1/n (15)

Figure 5 shows water saturation profiles for well A usingArchie’s parameters a, m and n calculated by fourtechniques. Water saturation was computed in clean sandsection with Archie’s formula while in shaly sand section itwas computed using shaly sand water saturation models.Water saturation has been measured in the laboratory on 10 core samples from the studied section in the well; theaverage water saturation of all core samples was 30.5%.Figure 5a illustrates water saturation profile with depth usingcommon values (a = 0.65, m = 2.15 and n = 2). Examinationof Figure 5a shows that water saturation is generally high andthe average water saturation for 85 points was 34.4% withstandard deviation (σSw

) values equals to 0.34. Figure 5b isthe water saturation profile versusdepth with use of Archie’sparameters calculated by conventional technique (a = 1.36, m = 2.03 and n = 2.1). The average water saturation of thestudied section of 85 points was 47% with standard deviation(σSw

) = 0.17. It is still high, but it is better than in the case of

646

0.008450 8550 8650 8750

Depth (ft)a) Common values techniques

8850 8950 9050

0.100.200.300.400.500.600.700.800.901.00

0.008450 8550 8650 8750

Depth (ft)b) Conventional technique

8850 8950 9050

0.100.200.300.400.500.600.700.800.901.00

0.008450 8550 8650 8750

Depth (ft)c) CAPE technique

8850 8950 9050

0.100.200.300.400.500.600.700.800.901.00

0.008450 8550 8650 8750

SW

SW

SW

SW

Depth (ft)d) 3D technique

8850 8950 9050

0.100.200.300.400.500.600.700.800.901.00

Figure 5

Water saturation profiles with different Archie’s parameters using four techniques for well A.

GM Hamada et al./ Water Saturation Computation from Laboratory, 3D Regression

common values. Figure 5c shows water saturation profilederived from water saturation equations using Archie’sparameters calculated by using the CAPE technique (a=3.21, m = 1.7 and n = 2.21). The average water saturation ofthe retained section was about 28% with standard deviation(σSw

) = 0.067. CAPE technique provided water saturationvalue much closer to the average value of core-measuredwater saturation (Sw core = 30.5%) than conventional andcommon techniques. Figure 5d depicts the water saturationprofile deduced from the use of Archie’s parameterscalculated by 3D technique (a = 2.94, m= 1.56 and n = 2.15).Average value of 85 water saturation points was found31.2%. With standard deviation (σSw

) equals to 0.069. Table5 summarises the average water saturation values computedby the Archie’s parameters deduced from the threetechniques. It is obvious that CAPE and 3D methods givewater saturation more accurate than the other methods andalso with lower standard deviation error. The 3D techniqueseems better than CAPE due to lower computing time and byits optimisation technique which is more physically

concerned with water saturation and related factors thanCAPE technique.

TABLE 5

Average water saturation and standard deviations using Archie’sparameters from four techniques for well A and well B

Well A

Technique Sw σw

Common values 34.4 0.34

Conventional 47 0.17

CAPE 28 0.067

3D 31.2 0.069

Well B

Technique Sw σw

Common values 10.2 0.38

Conventional 15.4 0.19

CAPE 6.5 0.072

3D 11.6 0.078

647

0.009450 9550 9650 9750

SW

Depth (ft)a) Common values techniques

9850 9950 10 050

0.100.200.300.400.500.600.700.800.901.00

0.009450 9550 9650 9750

SW

Depth (ft)b) Conventional technique

9850 9950 10 050

0.100.200.300.400.500.600.700.800.901.00

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Depth (ft)c) CAPE technique

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Depth (ft)d) 3D technique

9850 9950 10 050

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Figure 6

Water saturation profiles with different Archie’s parameters using four techniques for well B.

Oil & Gas Science and Technology – Rev. IFP, Vol. 57 (2002), No. 6

Figure 6 shows water saturation profiles for well Bdeduced from different water saturation equations (Archie’swater saturation equation in the case of clean sand and shalywater saturation equations in the case of shaly sand sections,10 core samples) measurements that give an average watersaturation of 12.25%. Figure 6a shows water saturationversusdepth, the Archie’s parameters used to compute watersaturation with the common values (a = 0.65, m = 2.15 and n = 2). Using 85 saturation points, the average water satura-tion was 10.5% with standard deviation (σSw

) = 0.38. Watersaturation profile with Archie’s parameters derived by theconventional technique (a = 0.95, m = 1.85 and n = 2.12) isshown in Figure 6b. The average value of an 85 saturationpoints of this curve was in the range of 15.4% with standarddeviation (σSw

) = 0.19. This reflects that water saturationcalculated by conventional is much better than that derivedfrom common Archie’s parameters. Figure 6c illustrates thewater saturation curve determined after the use of CAPEtechnique to Archie’s parameters (a= 2.44, m = 1.2 and n =1.95). Examination of this water saturation profile showsthat water saturation values are low with an average of 6.5%and standard deviation (σSw

) = 0.072. This average value ismuch less than core sample water saturation value. Figure 6dshows water saturation profile with the use of Archie’sparameters computed by 3D technique (a = 2.59, m = 1.43and n = 1.87). The average water saturation of 85 saturationpoints was found 11.6% and their standard of deviation was0.078. The average water saturation values using Archie’sparameters calculated by the different techniques arepresented in Table 5. It is clear that 3D technique gives betterstandard deviation and the closest average saturation to thecore samples measured water saturation value. Therefore it isrecommended to use the 3D technique which provides thevalues of Archie’s parameters a, m and n and with anaccepted water saturation error.

3 VARIABLE SATURATION EXPONENT AND WATERSATURATION

Laboratory measured saturation exponent (n) showed somevariations. An exact value of saturation exponent is necessaryfor a good log interpretation analysis aiming to a precisewater saturation determination. There are many factorsaffecting saturation exponent such as rock wettability, grainpattern, presence of certain authigenic clays, particularychamosite, which may promote oil wet characteristics andhistory of fluid displacement. However, it is found that rockwettability is the main factor affecting saturation exponent(n). Archie’s saturation equation makes three implicitassumptions: – the saturation/resistivity relation is unique;– n is constant for a given porous medium; – all brine contributes in the electric current flow.

It has been found [8] that these assumptions are valid onlyin water-wet reservoir. This is because the saturationexponent n depends on the distribution of the conductingphase in the porous medium and therefore depends on thewettability. The saturation exponent (n) is about 2 in water-wet rock, where brines spread over grain surface andfacilitate the flow of the electric current. While it may reach25 in strongly oil-wet rock, where oil coats grain surface andcauses disconnections and isolation of globules of brine andtherefore this brine will be unable to conduct a current flow.

TABLE 6

Change of water saturation with saturation exponent

for well A and well B

A - Common values techniqueWell A

n 2 4 6 8Sw 0.344 0.676 0.781 0.825

Well B

n 2 4 6 8Sw 0.154 0.367 0.510 0.601

B - 3D techniqueWell A

n 2.15 4 6 8Sw 0.312 0.641 0.745 0.811

Well B

n 1.87 4 6 8Sw 0.116 0.378 0.521 0.621

In order to test the effect of saturation exponent on watersaturation computation, parameters a and mwere fixed and ntook different values 2, 4, 6 and 8 in the case of commonvalues technique and values of 1.87, 2.15, 4, 6 and 8 in using3D technique for wells A and B. From experience, saturationexponents 1.87, 2, 2.15 correspond to water wet rockassumption and saturation exponents 4, 6 and 8 indicatepreferably oil-wet rock. Water saturation was computedusing the same 85 saturation points for well A and B. Thissensitivity test was carried out using Archie’s parameterscalculated by common values technique and 3D technique.The average water saturation values shown in Table 6 clearlyindicate the effect of saturation exponent on the watersaturation computation in wells A and B using commonvalues and 3D techniques. Figure 7a shows water saturationprofiles with variable saturation exponents using commonvalues technique and Figure 7b shows saturation profilesusing 3D technique for well A. Water saturation profiles withthe two techniques (common values and 3D) are shown inFigures 8a and 8b for well B. The comparison betweenFigures 7 and 5 for well A and Figures 8 and 6 for well Bdemonstrates clearly the increasing trend of water saturationprofiles as saturation exponent increases. These computed

648

GM Hamada et al./ Water Saturation Computation from Laboratory, 3D Regression 649

0.008450 8550 8650 8750

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Depth (ft)a) a = 0.65, m = 2.15, n = 2

8850 8950 9050

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Depth (ft)b) a = 0.65, m = 2.15, n = 4

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Depth (ft)c) a = 0.65, m = 2.15, n = 6

8850 8950 9050

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Depth (ft)d) a = 0.65, m = 2.15, n = 8

8850 8950 9050

0.100.200.300.400.500.600.700.800.901.00

Figure 7a

Change of water saturation with change of water saturation exponent (conventional technique) for well A.

0.008450 8550 8650 8750

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Depth (ft)a) a = 1.43, m = 1.97, n = 2.15

8850 8950 9050

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Depth (ft)b) a = 1.43, m = 1.97, n = 4

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Depth (ft)c) a = 1.43, m = 1.97, n = 6

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Depth (ft)d) a = 1.43, m = 1.97, n = 8

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Figure 7b

Change of water saturation with change of water saturation exponent (3D technique) for well A.

Oil & Gas Science and Technology – Rev. IFP, Vol. 57 (2002), No. 6650

0.00

SW

Depth (ft)a) a = 0.65, m = 2.15, n = 2

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Depth (ft)b) a = 0.65, m = 2.15, n = 4

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Depth (ft)c) a = 0.65, m = 2.15, n = 6

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0.00S

W

Depth (ft)d) a = 0.65, m = 2.15, n = 8

0.100.200.300.400.500.600.700.800.901.00

9450 9550 9650 9750 9850 9950 10 050 9450 9550 9650 9750 9850 9950 10 050

9450 9550 9650 9750 9850 9950 10 050 9450 9550 9650 9750 9850 9950 10 050

Figure 8a

Change of water saturation with change of water saturation exponent (conventional technique) for well B.

0.00

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Depth (ft)a) a = 0.92, m = 1.87, n = 1.87

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0.00

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Depth (ft)b) a = 0.92, m = 1.87, n = 4

0.100.200.300.400.500.600.700.800.901.00

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Depth (ft)c) a = 0.92, m = 1.87, n = 6

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Depth (ft)d) a = 0.92, m = 1.87, n = 8

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9450 9550 9650 9750 9850 9950 10 050 9450 9550 9650 9750 9850 9950 10 050

9450 9550 9650 9750 9850 9950 10 050

Figure 8b

Change of water saturation with change of water saturation exponent (3D technique) for well B.

GM Hamada et al./ Water Saturation Computation from Laboratory, 3D Regression

water saturations for well A and B showed an exponentialincrease of water saturation with the increase of saturationexponent. It may follow the following relations between theaverage water saturation and saturation exponent using allpoints in Table 6 of wells A and B.

Sw = 0.634 log 1.779n for 3D technique (16)

Sw = 0.794 log 1.171n for common values technique (17)

It is observed that the effect of saturation exponent onwater saturation computation is more significant at lowerwater saturation values than at higher values. From theaverage water saturation values in Table 6 and watersaturation profiles in Figures 7 and 8, we can conclude thatthe common values technique is more sensitive than the 3Dtechnique to changes in the saturation exponent. Thissensitivity reflects the exigency for an accurate saturationexponent determination to properly compute the watersaturation either by Archie’s formula or by the shaly watersaturation model.

CONCLUSIONS

Archie’s parameters have been determined by four techni-ques; common values, conventional, CAPE and 3D. Fromthe analysis and application of these techniques in thecomputation of water saturation either in clean sands or shalysands, the following conclusions may be drawn:– Conventional technique optimises the two functions F

versusφ and Rt vs. Sw rather than water saturation values.While CAPE technique confirms that the quantity to beoptimised is not the two functions but rather the watersaturation.

– The 3D technique provides simultaneously the values ofArchie’s parameters from standard resistivity measure-ments on core samples. Unlike the conventional method,which ignored the values of Sw < 1.0 in the determinationof aand m, the 3D method uses all data of Sw points.

– 3D technique answers the controversial question ofwhether tortuosity factor, a, should be fixed at unity ornot. It gives directly a, m and n, and thereby, it isrecommended to consider the case of the three variables a,mand n.

– For applications where the highest possible accuracy inhydrocarbon saturation is required, it is recommended touse the 3D technique, unless, there are adverse conditionsas mentioned in the text.

– The saturation exponent (n) has a great effect on watersaturation computation. The saturation exponent is

affected by many factors such as reservoir rock wettabilityand the presence of certain authogenic clays that maypromote oil wet characteristics. Therefore, it isrecommended to calculate the saturation exponentconsidering reservoir wettability conditions. On the side,there is no a significant effect of wettability alteration onthe value of formation resistivity factor, F.

REFERENCES

1 Archie, G.E. (1942) Electrical Resistivity Log as an Aid inDetermining Some Reservoir Characteristics.Trans., AIME,146, 54-62.

2 Atkins, ER. and Smith, G.H. (1961) The Significance ofParticle Shape in Formation Resistivity Factor-PorosityRelationship. JPT, March, 285-291.

3 Borai, A.M. (1987) A New Correlation for the CementationFactor in Low-Porosity Carbonates. SPE FormationEvaluation, December, 495-498.

4 Winsauer, W.O. and McCardell, W.M. (1954) Ionic-DoubleLayer Conductivity in Reservoir Rocks. Trans, AIME, 17-23.

5 Maute, R.E., Lyle, W.D. and Sprunt, E. (1992) ImprovedData-Analysis Method Determines Archie Parameters fromCore Data.JPT, January,103-107.

6 Ransom, R.C. (1984) A Contribution Toward a BetterUnderstanding of the Modified Archie Formation ResisivityFactor Relationship. The Log Analyst, March-April, 7-12.

7 Shouxiang, Ma. and Xiaoyun, Zh. (1994) Determination ofArchie’s Cementation Exponent from Capillary PressureMeasurements. Trans., Intr. Symposium on Well LoggingTechnology, X’ian, China, 83-103.

8 Sweeney, S.A. and Jenning, H.Y. (1960) The ElectricalResistivity of Preferentially Water-Wet and PreferentiallyOil-Wet Carbonate Rock,Producers Monthly, Schlumberger,24, 29-32.

9 Worthington, P.E. and Pallet, N. (1992) Effect of VariableSaturation Exponent on the Evaluation of HydrocarbonSaturation.SPE Formation Evaluation, December, 331-336.

10 Hamada, G.M., Assal, A.M. and Ali, M.A (1996) ImprovedTechnique to Determine Archie’s Parameters andConsequent Impact on the Exactness of HydrocarbonSaturation Values. SCA, Paper 9623, presented atIntl.Symposium of SCA, Sept. 8-10, Montpellier, France.

11 Doll, H.G. and Martin, M. (1954) How to Use Electric LogData to Determine Maximum Producible Oil Index. Oil andGas J., July, 77-82.

12 Poupon, A., Loy, M.E. and Tixier, M.P. (1954) AContribution to An Electrical Log Interpretation in ShalySands. JPT, June, 354-366.

13 Hossin, A. (1960) Calcul des saturations en eau par laméthode du ciment argileux (formule d’Archie généralisée).Bulletin AFTP, 125-128.

14 Dresser Atlas (1982) Well Logging and InterpretationTechniques,Dresser Atlas Ltd.

Final manuscript received in September 2002

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